Posterior Contraction Rates for the Bayesian Approach to Linear Ill-Posed Inverse Problems

TL;DR

This paper develops a Bayesian framework for linear inverse problems in Hilbert spaces, using PDE techniques to derive posterior contraction rates under weak prior and noise assumptions.

Contribution

It introduces a novel PDE-based method to analyze posterior contraction rates in Bayesian inverse problems with Gaussian priors and noise.

Findings

Derived explicit contraction rates for the posterior distribution.

Applicable to a broad class of inverse problems with weak prior-noise relations.

Utilized PDE techniques to handle unbounded precision operators.

Abstract

We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications.